Construction of quasi-periodic response solutions in forced strongly dissipative systems

TL;DR

This paper proves the existence of quasi-periodic response solutions in strongly dissipative forced systems with large damping, providing explicit estimates and a constructive method for solutions.

Contribution

It offers a fully constructive proof of response solutions in dissipative systems with explicit damping estimates, addressing a degenerate implicit function problem.

Findings

Existence of quasi-periodic solutions with the same frequency as forcing

Explicit bounds on minimal damping coefficient

Constructive method for solution determination

Abstract

We consider a class of ordinary differential equations describing one-dimensional quasiperiodically forced systems in the presence of large damping. We give a fully constructive proof of the existence of response solutions, that is quasi-periodic solutions which have the same frequency vector as the forcing. This requires dealing with a degenerate implicit function equation: we prove that the latter has a unique solution, which can be explicitly determined. As a by-product we obtain an explicit estimate of the minimal size of the damping coefficient.